Development of a hybrid,ANN-linear model for building drainage systems subject to multiple simultaneous discharges

Abstract

The discharge of wastewater from multiple floors into a vertical stack presents significant simulation challenges due to the complex two-phase flow, involving annular water movement and airflow induced by the ventilation network. These challenges are particularly pronounced in high-rise drainage systems. Previous studies by the authors successfully simulated classical pressure profiles using ANN models; however, limitations arose in predicting pressure variations under simultaneous discharge. In this study, a hybrid modelling approach was developed by integrating a linear component into the ANN framework. Initially, Feed Forward Back Propagation and Radial Basis Function ANN models were trained and tested. To enhance accuracy, a Particle Swarm Optimization (PSO) algorithm was later applied to optimize the Feed Forward model’s weights and biases. To capture different flow regimes, a hybrid ANN–Linear model was developed by separating dry-stack and wet-stack zone data. The predictions were validated against experimental data from a unique 32-storey drainage test facility (NLT Tower, UK). The model used discharge flow rate and stack height as inputs, with stack pressure as the output. The hybrid model, trained with real-world data, demonstrated reliable performance and proved an effective tool for designing and optimizing Building drainage system in high-rise buildings under simultaneous discharge conditions.

Practical Application

The simulation of two-phase flow phenomena in high-rise building with simultaneous discharges mimics the realistic situation. The hybrid ANN–linear model can be applied in the design and optimisation of high-rise building drainage systems to ensure safe and efficient operation under peak load conditions. By accurately predicting pressure profiles across both dry and wet stack zones, the model enables engineers to size vent pipes, select stack diameters, and configure branch connections to minimise the risk of trap seal loss, backflow, and system noise. Incorporating real-world data allows for site-specific calibration, supporting compliance with building codes and reducing overdesign.

Introduction

The two phase fluid flow phenomena within a vertical stack of a BDS is inherently unsteady. Variations in flow conditions within any fluid transport system can lead to localized pressure spikes, which appear as transient pressures or surges. In building drainage systems (BDS), these surges occur frequently due to the irregular discharge from sanitary fixtures, while the primary function of a BDS is to remove wastewater from a building. The pressure profile in different flow regime along the height of BDS is depicted. Its efficacy primarily relies on its ability to accommodate induced airflow. This airflow is induced by shear forces generated through the interaction between falling water in the main stack and the entrained central air core. ¹ In large and complex fluid systems, destructive pressure transients can propagate, potentially leading to system failure or necessitating the implementation of surge prevention measures during system design. The unsteady nature of fluid flow presents a significant risk to the structural integrity of BDS. Within the vertical stack, the flow is two-phase, with air being driven by the discharge of water. As these flow waves travel through the stack, they experience shear forces. ² Air pressure transients in drainage and vent systems (DVS) can compromise appliance trap seals, allowing unpleasant sewer gases to infiltrate indoor environments. Ultimately, rapid fluctuations in flow conditions are the primary cause of pressure transients. Extensive research, including studies by Gormley, ³ has examined the generation and propagation of air pressure transients and their impact on the air pressure regime within BDS. The variations in air pressure within the building drainage system which lead to induced siphonage where the propagation of the air pressure is predicted using differential equation method. ⁴ The pressure profile figure has been indicated in my previous paper. Comprehensive studies indicate that discharge from the stack descends in an annular form, inducing airflow due to pressure reduction. However, a significant portion of the water was also found within the air core. This research suggests that both the water annulus and the air core contribute to airflow entrainment. Considering these factors, this study investigates pressure profile in building drainage systems under simultaneous discharge conditions using ANN models and the reliability with respect to their applications has been found in FF-PSO and Hybrid linear-ANN models.

Overview of flow phenomena in BDS based on predictive modelling

Flow phenomena in BDS

The enhancement of fluid flow modeling is achieved by analyzing the interaction between wastewater and entrained airflow. ⁵ This study examines friction factor data and expands empirical datasets, improving system response predictions for flow within a vertical stack, whether with a single or multiple inlets. The analysis encompasses vertical stacks, branch connections, appliance traps, and sewer connections. The finite-difference method is employed to simulate unsteady flow, providing valuable insights into drainage network design. This approach aids in assessing the system’s time-dependent pressure response and predicting trap seal functionality. The report described an observational investigation of drainage sewer behaviour under hydraulic overloading. ⁶ The research investigates the behaviour of water and confined air in drainage systems experiencing surcharge and pressurized pipelines. This is achieved by analyzing the dynamics of air-water flow in a horizontal pipe being filled rapidly. In their work, patterns of air–water interface, air penetration and release of air in the pipe end are demonstrated through images. Synchronously, recorded pressure traces show how air–water phase development affects pressure oscillation patterns. The rapidly filled horizontal pipe demonstrates three distinct pressure oscillation behaviour at the air-water interface, each determined by the rate at which air is released from the orifice. The origins of positive air pressure transient propagation, demonstrated that these transients can be described by the St Venant equations of unsteady flow. When the entrained airflow channel is impeded, positive air pressure transients propagate at the local acoustic velocity, unaffected by air absolute pressure or pipe wall elasticity due to their low amplitude. Positive transients close air admittance valves and push trap seal water into connected appliances. The fluctuating water flow applied creates negative transient and transmits it throughout the network. Positive air pressure attenuation assessment could be made through the methods proposed. The experimental strategy to validate the concept, used the method of characteristics solution for the St Venant equations. This technique can considerably improve the designer’s capability to forecast flow phenomena while applying to both vertical stack flows with single and multiple inlets. When combined with the theory of irregular free surface flow in horizontal pipes, it permits the analysis of the entire drainage network. ⁷ In response to decelerating or accelerating flow conditions, transmission of air pressure transient in BDS may lead to either negative or positive pressure variations. Historically, negative transients were suppressed by local ventilation or utilizing inward relief air admission valves. Positive transients have only been addressed with open roof terminations. In this paper, a technique is introduced to mitigate positive air pressure transients by employing a flexible, variable-volume containment vessel. This vessel has the capability to decrease the rate of change of entrained air when a system experiences an overcharge. Following earlier research, air pressure in vertical stacks of a BDS is predicted empirically. An experimental system that replicates a medium-high flat provided empirical parameters and model verification. Average air pressure distribution can be predicted by using this model in a vertical stack under steady flow conditions with single-point discharge. ⁸ A set of experimentally determined formulas to explain the transport of a solid particle inside the vertical stack flow that depends on the Mach number which corresponds to the movement of the solid in the fluid medium. Waste solids in building drainage systems have been thought of as discrete objects that are susceptible to a variety of forces, such as frictional, hydrostatic, inertia and buoyancy forces. With the development of a measurement system based on an infrared pick-up device, the long-standing problems of detecting the velocity of the solids have been sorted out. ⁹ The design guidelines that protect these trap seals from unwelcome air pressures; nevertheless, they are based on steady-state tests using cold, pure water as a test medium and incorporate a significant “safety factor” without any formal justification. Employing a mathematical model (AIRNET), this work illustrates the influence of different parameters affected in the system while operating with wastewater being dosed with detergents over a range of temperatures. Further, assessment with respect to impact of the detergent on standards of the BDS design codes has been made. The findings show that present code-based estimates generally fall short, quantify the impacts of detergency with temperature, and raise the possibility that updated building rules may be necessary. ¹⁰ The Method of Characteristics (MOC), which applies to partial differential equations involving only two independent variables. He also noted that the MOC is commonly used in propagation problems, such as gas flow and water wave propagation, and is closely related to plasticity theory. ¹¹ Later, The MOC to unsteady, partially filled pipe flow problems using a finite difference scheme. They also introduced a numerical model that incorporates flow depth, velocity, and wave speed to simulate unsteady flow conditions in partially filled pipes. The AIRNET model, based on the continuity and momentum equations formulated through characteristic equations, requires significant computational time for evaluation. ¹² However, scarcity in the investigations that reports simultaneous discharge to main stack of BDS of a multi-story building (MSB) and consequent transport phenomena is noticed. Therefore, the present experimental investigation employs the realistic 32-storey building NLT tower test-rig for data accusation.

ANN model

Determination of the pressure distribution for air movement in the vertical stack from experimental investigation is very costly but reliable. The computational method, for designing an optimum strategy for the prediction of the pressure in the two-phase flow invertical pipe, an ANN model was opted. ¹³ The prediction of the friction factor and Nusselt number in a helical tube by using an ANN model is better in comparison to the other existing models. ¹⁴ An ANN model for predicting pressure drop in the horizontal pipe is reported for the multiphase flow. ¹⁵ The application of an artificial neural network (ANN) model with a back propagation (BP) learning algorithm is employed to predict the performance of the suction line outlet temperature and mass flow rate in heat exchanger in refrigeration systems. ¹⁶ The study examines two-phase flow phenomena in both horizontal and vertical tubes. A commercial computational tool, key characteristics such as pressure drop and different flow regimes were analyzed. Additionally, an ANN approach was implemented, with pressure drop as the output variable and flow velocity as the input parameter. The results revealed that numerical predictions were sensitive to inlet geometrical simplifications. Consequently, accurate modeling requires appropriate meshing and well-defined boundary conditions at the inlet. When comparing ANN predictions with experimental and numerical results, it was found that ANN did not outperform numerical simulations. The optimal results were obtained by testing various transfer functions and neuron configurations in the hidden layer. However, ANN-based simulations are recommended due to their efficacy and ease of execution. Despite being complex and time-consuming, ANN remains a viable method for predicting pressure loss in two-phase flow. ¹⁷ An ANN technique applied to a vertical pulsating heat pipe to address its heat transfer inside a pulsating heat pipe is found to be reliable. The need for the application of ANN techniques to pulsating heat pipe has emerged because of the lack available of reliable heat transfer correlations. ¹⁸ Particle Swarm Optimisation is opted for predicting non-linear functions. The computational optimisation performed by the PSO-ANN model is inexpensive in nature in terms of speed and memory. ¹⁹ Finding optimal space being the objective function while interacting with individual particles, the set of coefficients are determined which helps to control the convergence of the model. ²⁰ The position vector plays a vital role in improving the performance of the PSO Model. The position vectors are updated based on the second personal best and global best, which improves the PSO Model. ²¹ The ANN-PSO approach was employed to optimize the standard aeration efficiency of a Venturi aeration system, with aeration tests conducted in a tank to evaluate the effects of various operating variables. ²² The flow characteristics in the dry stack zone and wet stack zone found to be different and becomes challenging with respect to its ANN modeling. Furthermore, under simultaneous discharge conditions, it becomes cumbersome from modelling perspective. Therefore, all three ANN models such as FFBP, RBF and FF-PSO ANN models being developed by the author^23,24 are examined with respect to their reliability for simultaneous discharge boundary conditions. Ultimately, a hybrid linear-ANN model was explored to predict the pressure profile, whose algorithm will be detailed in the methodology section.

Methodology

It is understood that the structure of two-phase flow phenomena in the wet-stack is quite different from the phenomena in the dry-stack. The dry stack experiences the flow of entrained air because of the discharge flow received by the main stack at different locations. The viscous flow phenomena in the vertical stack happens to be the outcome of gravity force, traction force, interfacial shear stress and wall friction force inside the complicated geometrical configuration. The governing equations meant for two phase flow phenomena are nonlinear. As discussed, the pressure transients found to be the function of change in the velocity, flow properties and acoustic wave propagation. Further, resultant pressure transients do depend on the ventilation systems. The reflection and transmission of waves do affect it. In resolving such complex phenomena, researchers have attempted to find out pressure transients using Method of Characteristics and established some thumb rules for industrial practices. Researchers do contribute for improvement to AIRNET simulation software using Method of Characteristics. Further, researchers do resolve these problems using different numerical methods which are computationally expensive. So, experimental investigation of pressure transients in BDS is undertaken and the data obtained is used for ANN modelling. The description of experimental set-up and the details procedure is outlined in the following section Figure 1.

Figure 1.

Air pressure distribution in building drainage system.

Experimental set-up

The experimental setup for understanding the flow phenomena inside BDS at National lift tower, Northampton, UK is depicted through Figures 2(a) and (b). Figure 2(a) describes the single stack system with connections of different sources for discharge at different floors (F1, F2,….F15) into the main stack. The instrumentation for measuring pressure and flow rate has been provided. Provision of flow control is kept for simulating different realistic conditions of BDS in MSB. Further, main stack that has been supported with vented system is shown in Figure 2(a). Details of dimensions and arrangements are shown. The dump tank is installed at 14^th floor for providing steady water flow. Figure 2(b) shows the connections of eight pressure sensors (P1, P2…P8) at different locations of the vertical stack (i.e. 100 mm diameter pipe). There are three vent pipes attached to the vertical stack which are 50 mm, 75 mm and 100 mm in diameter as indicated in Figure 2(b). There is provision of both closing and opening of ventilation pipes, while conducting experiments. The pressure sensor P1 and P2 located in the vertical stack in such a way that the pressure sensor P1 is connected at 1  m upstream of the collection tank whereas the P2 is connected at 1  m downstream of the collection tank means P1 and P2 are in the height located at 0.0 m and 0.9 m respectively.

Figure 2.

Pressure sensors installation in NLT Tower (a) Schematic diagram of NLT Tower (b) Real image of Pressure sensors installation.

The pressure sensor P3 is placed just below the T junction of the toilet which is located at 8.2 m height. The pressure sensor P4 is placed 1 m above the toilet is placed i.e. at height of 10  m. The pressure sensor P5 is located at height of 0.1 m above the P4. The pressure sensor P6 is at 27.9 m which is also located at just one m above the toilet. This location is also just above the T-junction of the toilet which connects to the secondary vent line. Similarly, the pressure sensor P7 is connected above the toilet and just above the T-junction of the toilet which is connected to the secondary vent line. The pressure sensor P8 is situated in 15^th floor, i.e. located at just one m above the toilet and just above the T-junction of the toilet. This connects to the secondary vent line which is placed at height of 75.6 m. The pressure sensors are maintained in the range of 1 bar. The anemometer placed at the top of the tower (i.e. at height of 73.6 m) is used to measure the air flow rate into the vertical stack. Figure 2(a) shows the recirculation of water through the provision of a pump and dump tank. The sensors are connected to data acquisition system to record both water and air flow rates and pressures at different locations along the vertical stack. This arrangement has been made to record the readings of pressure sensors along the vertical stack for both transient and steady state condition. Provision has been made to vary the location and flow rate of discharge. In the present experimental work, two cases of steady flow rates i.e. 1.8 L/s and 1 L/s, have been investigated considering all opened ventilation pipes. This section explains about the experimental set-up at NLT tower. These experiments conducted on NLT tower are based on different configurations such as discharge flow taking place from different discharge floor height for the steady flow conditions Experiments are conducted with simultaneous discharge with different discharge flow rate. The pressure measurements are also considered for the set of data operating under steady and flush for different discharge floor height with different discharge flow rate. These data collected helped for analyzing the flow phenomena and further used as the target data for ANN modelling.

ANN model

In FFNNs, the information is passed in one direction, i.e. the information moves from input layer to the output layer inside the network. It does not form any loop or cycle. In absence of the target data, comparison of the predicted data cannot be made to estimate the error associated with the model. When the target data is available against the predicted data, the principles of supervised learning can be implemented. FFBP-ANN model happens to be a dynamic NN, whose functionality is depicted in Figure 3. The universal approximation theorem of Neural Networks states that a standard multi-layer feed forward network with a single hidden layer with arbitrary activation function is a universal approximator. ²⁵

Figure 3.

Architecture of dynamic NN.

The network shown above through the Figure 3 represents the connection between the input neuron and the output neuron. So, the topology explained in Figure 3 shows the arrangement of neurons in the in the input layer (I), hidden layer (H), and output layer (O) that are interconnected through weight vectors, bias and transfer functions. Number of neurons in the input layer (I), hidden layer (H), and output layer(O) are M, N and P respectively.

The linear transfer function, tan-Sigmoid transfer function and log-Sigmoid transfer function are considered to be the activation functions at input layer, hidden layer and output layer respectively. The input and output from a neuron are depicted by ‘I’ and ‘O’ respectively through 1^st subscript.

This section describes the FF-PSO ANN model for modeling two-phase flow phenomena in vertical stack of BDS.^23,24 The output from i-th neuron at the input layer (I) considering linear transfer function is represented as:

${\hspace{0.17em}I}_{Oi}={\hspace{0.17em}I}_{Ii}$

(1)

Where i = 1,2,3……….M.

The inputs to the hidden neurons ( $H_{Ij})$ that are connected to neurons at input layer through weight vector W_i,j and bias b₁, can be represented as:

$H_{Ij}=W_{1j}{I}_{O1}+\dots +W_{ij}{I}_{Oi}+\dots +W_{Mj}{I}_{OM}+b_1$

(2)

Where j = 1,2….N

The use of the log-sigmoid function with $a_1$ coefficient for hidden neurons, yields output from the hidden neuron ( $H_{Oj})$ as:

${\hspace{0.17em}H}_{Oj}=\frac{1}{1+e^{-a1H_{Ij}}}$

(3)

The input to the output neuron ( $O_{Ik})$ can be determined using V( $V_{1k}$ …. $V_{Nk}$ ) weight vector and bias b_2, as :

$O_{Ik}=V_{1k}{H}_{O1}+\dots +V_{jk}{H}_{Oj}+\dots +V_{Nk}{I}_{ON}+b_2$

(4)

The output from the output neuron uses the tan-sigmoid function with a₂ coefficient and is denoted as $O_{Ok}$ , which can be represented as

${\hspace{0.5em}O}_{Ok}=\frac{e^{a2O_{Ik}}-e^{-a2O_{Ik}}}{{\hspace{0.17em}e}^{a1O_{Ik}}+e^{-a1O_{Ik}}}$

(5)

The tan-sigmoid activation function at the hidden inlayer and the log-sigmoid function at the output layer is preferred. ²⁶ The mathematically defined error $(E_k$ ) as the representation of difference between the target data ( $T_{Ok}$ ) and predicted data ( $O_{Ok})$ from the output layer as:

$E_k=\frac{\left(T_{Ok}-O_{Ok}\right)^\left(2\right)}{2}$

(6)

The steps followed through equations (1) to (6), outlines the mathematical representation of Feed Forward Neural Network (FFNN) algorithm with weight vectors, bias and activation functions. The FFNN with guessed weight vector, bias and chosen activation function yields the prediction and when compared against the target data gives an idea about error. The error thus resulted can be minimized using FFBP ANN algorithm by updating guessed weights which has been already discussed in the previous work of the author. ²³ The same algorithm is employed in the present work for simultaneous discharges received by the vertical stack. Further, RBF-ANN model meant for modeling two-phase flow phenomena, that has been outlined in the work, ²³ has also been employed for applying to simultaneous discharge conditions in the present work. The back propagation for reducing error employing descent gradient method has been substituted by Particle Swarm Optimization (PSO) natural algorithm to establish its suitability.

Particle Swarm Optimization (PSO)

PSO evolutionary algorithm has been described in the following, is applied to simultaneous discharge condition for the present work. As the ANN schemes are not deterministic with respect to their better performance to predict characteristics of two-phase flow nonlinear phenomena, attempt is made to examine their suitability.

In PSO, ²⁴ the input data (i.e. floor height ‘FL’ better performance discharge flow rate ‘Q_W’) as well as the output data (i.e. Pressure ‘P’) are normalized as per the following and scaled between the value 0 to 1

${FL}_{norm}=\frac{FL-{FL}_{\mathit{min}}}{{FL}_{\mathit{max}}-{FL}_{\mathit{min}}}$

(7)

Similarly,

$Q_{wnorm}=\frac{Q_w-Q_{wmin}}{Q_{wmax}-Q_{wmin}}$

(8)

$P_{norm}=\frac{P-P_{\mathit{min}}}{P_{\mathit{max}}-P_{\mathit{min}}}$

(9)

After normalizing the data, the original data is segregated into two parts, which are to be used for testing and training purpose. Out of the whole data, 80% of the data is used for training whereas 20% of the data are used for testing. However, the errors which back propagated for training of data, updates the weights and bias to minimise the error through PSO-evolutionary algorithm.

Each particle in the network, while searching in the multi-dimensional space shown through Figure 4, has to modify itself according to its position and position of other particle in the swarm. The population algorithms are assigned with the random velocity and are allowed to move into different problem space. The particle has been assigned with the best position which has greatest fitness which is known as the global best position of the swarm. Each particle is assigned with some random velocity while moving in the problem space. The particle of the i-th position in the problem space is represented as:

$X_i=\left(x_{i1}+x_{i2}+\dots \dots \dots \dots \dots x_{id}\right)$

(10)

Where d is the dimension in the problem space.

Figure 4.

Component Vectors in PSO problem space. ³⁵

Similarly, the velocity of i-th position of the particle is denoted by

$V_i=\left({\mathrm{V}}_{\mathrm{i}1}+{\mathrm{V}}_{\mathrm{i}2}+\dots \dots \dots \dots \dots {\mathrm{V}}_{\text{id}}\right)$

(11)

The best position of the i-th particle can be denoted as

$P_i=\left(p_{i1}+p_{i2}+\dots \dots \dots \dots \dots p_{id}\right)$

(12)

Also, the global best position can be represented as

$P_g=\left(p_{g1}+p_{g2}+\dots \dots \dots \dots \dots p_{gd}\right)$

(13)

The velocity and position of the particle are updated and represented as

$V_{id}\left(t+1\right)=\mathrm{w}{V}_{id}\left(t\right)+C_1{r}_1\left(P_{i1}-X_{id}\left(t\right)\right)+C_2{r}_2\left(P_{gd}-X_{id}\left(t\right)\right)$

(14)

${\mathrm{X}\hspace{0.17em}}_{\text{id}}\left(\mathrm{t}+1\right)={\mathrm{X}\hspace{0.17em}}_{\text{id}}\left(\mathrm{t}\right)+{\mathrm{V}}_{\text{id}}\left(\mathrm{t}+1\right)$

(15)

Both velocity and position of the particle at (t + 1)th iteration are noted as $V_{id}\left(t+1\right)$ and ${\mathrm{X}\hspace{0.17em}}_{\text{id}}\left(\mathrm{t}+1\right)$ respectively. Equation (14) represents the update of the velocity of the particle which consists of three components such as momentum part, cognitive part and social part.

Here, w $V_{id}$ serves as a momentum part which indicate about the memory of the previous flight and prevent the particle from changing its direction denoted as 1^st component. They are biased according to the previous direction. $C_1{r}_1\left(P_{i1}-X_{id}\left(t\right)\right)$ defines as the cognitive part which describes about the performance of the i-th particle with respect to the past performance and particles are drawn back to their own position. The cognitive part is describing as 2^nd component. The social part $C_2{r}_2\left(P_{gd}{-X}_{id}\left(t\right)\right)$ indicates the performance relative to the neighbours which indicate the difference between the global best particle as well as the position of the particle which is denoted as third component The figure represented above describes about the components used in equation.

The momentum part which consists of inertia weight (w) plays a very important role in the problem space as shown through equation (16). The movement of the particle in the space is influenced through inertia weight (w) to consider the effect of the previous velocity. The higher inertia weight w speaks about the exploration term and smaller inertia weight ‘w’ helps in controlling the exploitation in the search space indicating about tuning the current search space. Therefore, inertia weight (w) is an important factor as it creates a balance between exploration and exploitation in the search space. ²⁷ The decreasing of inertia weight from 0.9 to 0.4 linearly with the changing iterations ²⁸ is reported. The inertia weight varies with the number of the iterations. For improving the convergence performance, the inertia weight (w) is modified by implementing damping ratio ‘α’. ²⁹ The modified inertia weight(w) was adopted for each iteration about the dynamic value of the inertia weight ³⁰ which can be represented as:

$\mathrm{w}’=\mathrm{w}*\mathrm{u}-\mathrm{k}\hspace{0.17em}\text{where}\hspace{0.17em}\mathrm{w}€\left[0,1\right],\mathrm{u}€\left[1.0001,1.005\right]$

(16)

where, w’ describes about the dynamic weight whereas w indicates about inertia weight whose value lies in between 0 and 1 and u lies in between the value 1.0001 and 1.005. The initialisation of the iteration (i.e k) takes place from the k = 0. Here the damping coefficient is defined as u^-k. The damping ratio is multiplied in every iteration.

The update of inertia weight with respect to iteration, ³¹ can be expressed as:

$\mathrm{w}=w_{\mathit{max}}-\frac{\left(w_{\mathit{max}}-w_{\mathit{min}}\right)t}{t_{\mathit{max}}}$

(17)

where the $w_{\mathit{max}}$ is defined as the initial values of the inertia weight vector and $w_{\mathit{min}}$ is the final value of the inertia weight vector. The term ‘t’ indentifies as the number of iteration whereas the $t_{\mathit{max}}$ defined as the maximum number of the iterations. So, the inertia weight can be adopted in a form as represented in 17. A constriction factor to equation (18) for improving the performance of the algorithm is included thereby making the solution scheme ³² more stable. The velocity equation in the search space can be reformed as:

$V_{id}=\chi *\left[V_{id}+C_1{r}_1\left(P_{i1}-X_{id}\left(t\right)\right)+C_2{r}_2\left(P_{gd}-X_{id}\left(t\right)\right)\right]$

(18)

Where

$\chi =\frac{2k}{|2-\phi -\sqrt{{\phi }^2-4\phi }|},\phi =C_1+C_2,\phi >4$

(19)

$\chi$ is the constriction factor which is employed to enhance the convergence speed of the PSO algorithm. ³² The term k controls the exploration and exploitation of the swarm which varies within 0 to 1. So, if k = 1, the convergence is very slow with the prediction of the exploration and if ‘k’ is near to 1, the convergence value is very faster with a value of exploitation. ³²

Shi (2001) also indicated the velocity clamping factor where V_max plays as a controlling factor for the exploration in the swarm. If V_max is larger, it facilitates the global exploration while small value of V_max, it encourages the exploitation. Higher value of the V_max can encourage exploration but sometimes the particles may miss the search space. ³³ The smaller value of V_max encourage exploitation which may get stuck at local optima. The value of Velocity clamping is expressed as:

$V_{\mathit{max}}=\mathrm{\delta }\left(X_{\mathit{max}}-X_{\mathit{min}}\right)$

(20)

Where X_max and X_min are the maximum and minimum value of the particle position and δ value is considered between 0 and 1. ³³ The performance of the PSO algorithm is better in the problem search space when swarm size operates within the range 20-50. ³⁴ This scheme has been employed to refine the weight vectors and helps to obtain optimal solutions by reducing the error.

In summary, to evaluate the effectiveness of the developed model, its performance was compared with that of the RBF, FFBP, and FF-PSO-ANN models under simultaneous discharge conditions. All these models utilize input variables such as discharge floor height and discharge flow rate, which are fed into the input layer neurons, processed through hidden layer neurons and activation functions, and finally aggregated at the output layer. The overall performance of each model is strongly influenced by the choice of activation function, the type of ANN model employed, and the number of neurons in the hidden layers.

Hybrid linear-ANN model

Further an attempt is made to develop a hybrid model for predicting the pressure profile of both dry-stack region and wet-stack zone. As observed from the literature, the pressure variation in the dry-stack appears to be linear whereas the pressure profile in the wet-stack zone does not fit into any mathematical non-linear functions or their combinations. Therefore, in the present work a linear model has been applied only to dry-stack domain and FF-PSO-ANN model has been employed for the wet-stack zone. Thus, the hybrid method employed in the present work happens to be a combination of the linear dry-stack model and FF-PSO-ANN model for wet-stack zone. The flow chart for the Hybrid linear-ANN model is given below Figure 5.

Figure 5.

Algorithm of hybrid Linear-ANN model.

In response to this, the current section addresses the segregation and modelling of data specific to the dry-stack and wet-stack zones. Considering the distinct characteristics of the dry-stack and wet-stack zones, a linear model is employed for the former, while a FF-PSO-ANN model is utilized for the latter. The hybrid model, thus combining a linear approach for the dry-stack and an ANN model for the wet-stack, is developed and thoroughly discussed in terms of its performance. The equation representing the linear model for the dry-stack zone is given by:

$\mathrm{P}=K*(\text{mH}+\mathrm{C})$

(21)

where m = $\frac{1}{2.147}$ and C = − $\frac{H_t}{2.147}$

Here, $H_t$ denotes the total height of the building, and H represents a specific height at which pressure is to be predicted. The value for ‘K’ has been found for all cases of BDS considering it as a fixed constant and varying constant with H. The linear model for the dry stack closely aligns with Darcy’s equation. Equation (21) expresses through linear model, being modified by a variable coefficient K. When the pipe receives a discharge, a negative pressure is generated, which draws ambient air into the system. The resulting air movement makes the Darcy relationship non-linear. To capture this behavior, the flow is modeled using a linear equation coupled with a variable constant K, whose value is determined empirically within the investigated range. The friction pressure loss is proportional to the length of pipe in the dry-stack zone as it is subjected to low air flow.

${\mathrm{\Delta }P}_f=\left(\frac{{-4\rho fLV}^2}{2D}\right)$

(22)

Results and discussion

In this section, obtained experimental data from the 32-story building test rig (NLT, UK) is analyzed and used as the target data for establishment of a suitable ANN model for the BDS receiving simultaneous discharges from different floors. Author has developed three ANN models^23,24 such as RBF-ANN model, FFBP-ANN model and FF-PSO ANN models for addressing two phase flow phenomena inside BDS for single discharge conditions.

In the present work, author has applied all these three ANN models to predict pressure profile in the BDS receiving simultaneous discharges from different floors. The performance of these models are evaluated and need for further accuracy is felt. Further improvement through development of a hybrid model is also reported. Prior to discussion on the pressure profile under simultaneous discharge conditions, validation of developed hybrid model with experimental result and FF-PSO ANN model is conducted for single discharge condition in the following section.

Validation with hybrid linear-ANN model

The Two-phase flow phenomena is observed in the wet stack zone whereas single phase phenomena is realized dry stack zone. Challenge remains to have a single ANN model to predict pressure profile for the whole height of the vertical stack of the BDS. The prediction using Hybrid model and FF-PSO ANN model has been validated against the experimental work of.

Cheng et al. ⁸ for a discharge coming either from 9^th or 10^th floor of a 13-story 40m height BDS. Further, the flow rate has been varied (1 L/s and 2 L/s) and the reliability of both models are tested. The results are depicted through Figure 6 and Table 1.

Figure 6.

Comparison of Hybrid-Linear Model with FF-PSO-ANN model (a) discharge flow from 10^th floor at 1 L/s (b) discharge flow from 9^th floor at 1 L/s (c) discharge flow from 9^th floor at 2 L/s.

Table 1.

Comparison of Hybrid linear ANN model, when discharge comes from a single floor.

Discharge flow rate	Discharge floor height (m)	Hybrid model					FF-PSO-ANN model
		K-constant		K (varying) = ${\hspace{0.5em}\mathrm{K}}_0\mathrm{H}+{\mathrm{K}}_1$			MRD (%)
		K	MRD (%)	${\hspace{0.5em}\mathrm{K}}_0$	${\mathrm{K}}_1$	MRD (%)
1 L/s	10th floor	1.63	24.99	0.17	7.12	13.13	20.56
1 L/s	9th floor	1.01	13.69	0	1.01	13.69	16.69
2 L/s	9th floor	1.12	20.18	0.00	1.13	12.24	18.43

The results produced by the hybrid linear ANN model were compared to those obtained from the FF-PSO-ANN model. Despite the FF-PSO-ANN model demonstrating superior performance compared to previously mentioned models,^23,24 it somewhat introduces a swirl in the dry-stack zone. Consequently, a linearized model in alignment with Darcy’s equation is introduced in the methodology section. Figure 6 depicts the comparison between the hybrid linear ANN model and the FF-PSO-ANN model. It is observed that the predictions from hybrid linear ANN model when compared with experimental data of Cheng et al. ⁸ is found to be relatively accurate compared to the FF-PSO-ANN model, as evident in the dry-stack zone shown through Figure 6.

The agreement of prediction of the pressure profile of FF-PSO and Hybrid linear-ANN model for discharge of 1 L/s from 9^th floor is very good with experimental results (Figure 6(b)). When the discharge increases to 2 L/s, some disagreement is observed with FF-PSO model in the dry-stack region (refer Figure 6(c)). Similarly, when discharge comes from 10^th floor with 1 L/s, some deviation in pressure profile is observed in the dry stack zone with FF-PSO model (Figure 6(a)). Even though FF-PSO happens to be a reliable model for non-linear phenomenon, but dry stack region and wet-stack region experiences different flow characteristics. So, the response of the hybrid linear-ANN model is assessed and found to be more accurate in comparison to FF-PSO model. The linear characteristics of the dry-stack zone are also attempted through fixed constant (K) and variable constant (K) whose effect is presented in Table 1.

Remarkably, the error is minimized and found to be less than FF-PSO model, when K is treated as a varying constant, as observed from Table 1.

Performance of ANN models for single discharge

Furthermore, experiments are conducted for the open system for two discharge rates i.e., 1.8 L/s and 1 L/s coming from 9^th floor of NLT tower test-rig.. Thus, the pressure profile for the BDS obtained is shown through Figure 7. The assessment of predictions from all three models are made through performance parameters given in Tables 2 and 3. Deviation in the dry-stack regime is seen mostly in the predictions of the RBF model. This prompted to predict the pressure profile through hybrid linear-ANN model and the improvement in the pressure profile is very much realized (Figure 7). The correlation coefficient and relative deviation confirms better predictions of both FFBP ANN and FF-PSO ANN model compared to RBF ANN model. The performance parameters for hybrid linear-ANN model indicate its further better accuracy and justify its development.

Figure 7.

Discharge from 9^th floor at 1.8 L/s using hybrid linear-ANN model.

Table 2.

Performance of three Models for NLT tower data for discharge at (1 L/s).

Discharge floor	Model	MSE	${\text{R}}^{\mathbf{2}}$	MRD (%)
9th floor	FF- PSO ANN	1.3874e-04	0.9984	1.81
9th floor	FFBP-ANN	2.4873e-04	0.9970	2.65
9th floor	RBF-ANN	0.0030	0.9791	3.70
9th floor	Hybrid Model	1.1874e-04	0.9995	1.51

Table 3.

Performance of three model for NLT Tower data for discharge 1.8 L/s.

Discharge floor	Model	MSE	${\text{R}}^{\mathbf{2}}$	MRD (%)
9th floor	FF- PSO ANN	4.8553e-05	0.9982	2.18
9th floor	FFBP-ANN	1.3874e-04	0.9894	2.65
9th floor	RBF-ANN	0.0028	0.9794	5.69
9th floor	Hybrid Model	2.8553e-05	0.9991	1.81

Performance of ANN models in simultaneous discharge

Pressure profile along the height for the two-phase flow phenomena in BDS of a 32-storeyed building when simultaneous discharge with steady flow rate of 1.8 L/s from both 9^th floor and 14^th floor enters into main stack is experimentally obtained by closing the ventilation pipe and opening ventilation system. These experimentally obtained data is used as the target data for the ANN models.

Single stack with simultaneous discharge

The predicted pressure profiles from all ANN models, alongside the experimental data, are illustrated in Figure 8(a). A comparison with this pressure profile with that of Figure 7(a), reveals that peak positive pressures are significantly higher under simultaneous discharge conditions. RBF-ANN model predictions are embedded with spurious oscillations. Other two models such as FFBP and FF-PSO ANN models experience some deviation in the dry-stack regime. However, predicted pressures show strong agreement even in the dry stack regime for the Hybrid linear-ANN model. The scatter plot (Figure 8(b)) for the hybrid linear ANN model confirms its better performance and reliability for employment.

Figure 8.

Simultaneous discharge from 9th and 14th floor with 1.8 L/s for single stack system (a) pressure profile (b) Scatter plot.

As observed from Table 4, the FFBP and FF-PSO ANN models accurately match the experimental data, while the RBF ANN model exhibits a maximum deviation of 10.48%. This performance of RBF-ANN model makes its use questionable for prediction of pressure profile under simultaneous discharge condition. The hybrid model demonstrates the best performance, achieving the lowest relative deviation (RD) of 1.02% and the highest R² value of 0.9984.

Table 4.

Performance of three ANN Models for discharge of 1.8 L/s coming simultaneously from 9^th and 14^th floor for single-stack system.

Model	MSE	${\mathrm{R}}^2$	MRD (%)
FFBP ANN	6.2297e-04	0.9975	4.96
RBF ANN	0 .0015	0.9939	10.48
FF-PSO ANN	8.6198e-04	0.9981	1.48
Hybrid model	9.8798e-04	0.9984	1.02

Discharge flow from different location simultaneously

The experiments are conducted for the steady case where the discharge flow rate takes place only from 9^th floor and as well as 14^th floor. Experiment has been repeated in the same test rig by opening the stack diameter of 50 mm ventilation pipe for the same steady simultaneous discharge conditions. Pressure profiles are shown through Figure 9 and the performance indicators are listed in Table 5. By comparing pressure profile given through Figure 9, both the peak negative pressure and peak positive pressure do reduce in their magnitude in single stack with ventilation system. Small deviations in agreement are observed in dry stack regime. FF-PSO ANN model seems to the better model among the three and RBF ANN model is prone to higher relative deviation. The improvement in the pressure profile by induction of hybrid model is observed through Figure 8(b) and from performance parameter i.e. highest correlation coefficient and least MRD. The noticeable oscillatory deviation in the dry-stack regime of RBF-ANN model and small deviation of the FFBP and FF-PSO ANN model is not seen in the Hybrid model. The performance of Hybrid model seems to be well acceptable for simultaneous discharge conditions. It gives least MRD of 1.07%.

Figure 9.

Pressure Profile for Simultaneous Discharge from 9th and 14th Floors with secondary ventilation 50 mm diameter pipe.

Table 5.

Performance of three ANN models for a simultaneous discharge of 1.8 L/s from the 9th and 14th floors.

Model	MSE	${\mathrm{R}}^2$	MRD (%)
FFBP-ANN	0.0011	0.9895	5.52
RBF-ANN	0.0017	0.9733	9.21
FF-PSO-ANN	7.7184e-04	0.9982	1.33
Hybrid Model	3.7184e-04	0.9992	1.07

This hybrid linear ANN model is found to be suitable for single stack with ventilation and without ventilation. Furthermore, it is found to be suitable for single stack being connected with secondary ventilation pipes for simultaneous discharge condition. However the simulations for simultaneous discharge condition received by single stack for varying location and varying pipe diameter may location of the discharge and the different stack diameter. The limitation of FF-PSO-ANN such as the large sample size is a requirement for better prediction.

Further, experiments have been conducted in the same test rig by opening the ventilation pipe of 75 mm diameter for the steady simultaneous discharge conditions and closing other two ventilation pipes. Both the predicted pressure profiles from ANN models and experiments have been shown in Figure 10 and the performance indicators are listed in Table 6. Figure 10 reveals that the spurious oscillatory pressure profile in the dry-stack portion of RBF-ANN model is unsuitable under simultaneous discharge conditions, which prompts for implementing hybrid model. FFBP-ANN represents dry stack and wet-stack pressures well but shows divergence at the positive pressure zone at the base of the stack.

Figure 10.

Pressure Profile for Simultaneous Discharge from 9th and 14th Floors by opening the secondary pipe of diameter 75 mm.

Table 6.

Performance of ANN models for discharge of 1.8 L/s coming simultaneously from 9^th and 14^th by opening the secondary pipe of diameter 50 mm.

Model	MSE	${\text{R}}^{\mathbf{2}}$	MRD (%)
FFBP ANN	6.2265e-04	0.9894	5.84
RBF ANN	1.8938e-04	0.9792	7.83
FF-PSO ANN	1.7754e-04	0.9981	1.21
Hybrid Model	1.2554e-04	0.9993	1.13

By comparing pressure profile given through Figure 10, both the peak negative pressure and peak positive pressure do reduce in their magnitude in ventilation system. FF-PSO ANN model seems to be the best model among the three and RBF ANN model is prone to higher relative deviation of 7.83%. The evolutionary algorithm used in FF-PSO found to yield higher accuracy. Further, this accuracy has been enhanced by employing hybrid model as evident from its statistical parameters given in Table 6.

In simultaneous discharge condition, the fluid comes out with positive gauge pressure whereas in single discharge condition, it comes out with almost zero-gauge pressure. It has been noted that the gage pressure value drops as the ventilation opening’s diameter rises. In this case spurious oscillation exhibited by in the dry-stack regime of RBF-ANN model is reduced. However, statistical parameters indicate that RBF-ANN model yields poor performance and Hybrid linear-ANN model exhibits the best. Statistical performance indicators (Table 6) helps to infer that hybrid model development for simultaneous discharge condition is justified by closing or opening the secondary pipe. It is reliable with respect to its use.

As observed from the results, it can be noticed from the pressure profile that the pressure variation characteristics distinctly divides the flow phenomena into two regimes that is dry-stack zone and wet-stack zone. The dry-stack regime is found with linear variation of the pressure while the wet-stack with single-phase flow whereas wet-stack zone comes across the non-linear variation for the two-phase flow (air and water). The single ANN model for the which emphasis the flow phenomena in the wet-stack fells to predict in the dry-stack zone for RBF-ANN model for simultaneous discharge condition. This justifies the development the hybrid linear FF-PSO-ANN model which yields better results as compared to other three models.

Both Figure 11 and Table 7 depict predicted pressure profile and performances of ANN models for simultaneous discharge of 1.8 L/s coming from 9^th and 14^th floor while keeping only 100 mm ventilation pipe open. Comparing the pressure profiles, given it is observed that both peak negative pressure and peak positive pressure is further reduced for single stack and with ventilation system when 100 mm pipe remains open. The dry stack regime comes across with small deviations in predicted pressure in open ventilation system when compared to closed ventilation system, it becomes minimum when 75 mm ventilation pipe remains open. In case of simultaneous discharge condition, the FF-PSO model performance is subjected to least MRD and high R² value. RBF ANN for this case yield MRD 0f 11.14% which is represented in Figure 11 and Table 7.

Figure 11.

Pressure Profile for Simultaneous Discharge from 9th and 14th Floors by opening the secondary pipe of diameter 100 mm.

Table 7.

Performance of Three ANN Models for discharge of 1.8 L/s coming simultaneously from 9^th and 14^th floor by opening the secondary vent pipe of diameter 100 mm.

Model type	MSE	${\text{R}}^{\mathbf{2}}$	MRD (%)
FFBP ANN	4.2752e-04	0.9983	5.54
RBF-ANN	0.0011	0.9954	11.14
FF-PSO ANN	0.0013	0.9987	1.49
Hybrid Model	6.752e-04	0.9989	1.26

Conclusion

Based on this investigation, the following conclusions can be drawn regarding the two-phase fluid flow phenomenon in a building drainage system under simultaneous discharge conditions. Additionally, the linear hybrid ANN model successfully captured these flow characteristics and demonstrated strong potential as a reliable tool for performance prediction in such systems.

The development of Hybrid linear ANN model as a refinement model over FF-PSO-ANN model found to be justified. Hybrid Linear ANN model exhibits higher value of R², lower value of relative deviation and MSE than FF-PSO-ANN, FFBP and RBF ANN models. The performance of RBF-ANN model is relatively unsuitable for its employment under simultaneous discharge condition.

Hybridization of a linear dry-stack model and FF-PSO-ANN model for wet-stack zone has been validated for pressure profile predicting satisfactorily against experimental data. This model is found to be more accurate than the other three ANN models. The concept of this segregation of the data for the dry-stack and wet-stack zone for training and testing of ANN models is more descriptive as flow features in these domains are quite different as such have led to better performance of Hybrid model. The hybrid linear-ANN model performs more effectively when simultaneous discharge occurs from different locations, both in the single stack configuration and in the single stack with ventilation.

Other observations from this study show that peak positive pressures are greater under simultaneous steady discharge conditions than under single steady discharge conditions. Furthermore, the single stack configuration experiences higher peak pressures for both positive and negative, than the stack with ventilation.

The results also show that in a single stack, increasing the ventilation pipe’s diameter results in lower peak positive and peak negative pressures.

Overall, the hybrid linear ANN model has been shown to be more descriptive of the observed performance of building drainage systems and could form the basis for an effective way to model BDS system performance from fundamental data associated with a design.